Properties

Label 2-28e2-7.4-c1-0-11
Degree $2$
Conductor $784$
Sign $0.605 + 0.795i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 2.59i)3-s + (−0.5 − 0.866i)5-s + (−3 − 5.19i)9-s + (−0.5 + 0.866i)11-s − 2·13-s + 3·15-s + (1.5 − 2.59i)17-s + (−2.5 − 4.33i)19-s + (−1.5 − 2.59i)23-s + (2 − 3.46i)25-s + 9·27-s − 6·29-s + (0.5 − 0.866i)31-s + (−1.5 − 2.59i)33-s + (2.5 + 4.33i)37-s + ⋯
L(s)  = 1  + (−0.866 + 1.49i)3-s + (−0.223 − 0.387i)5-s + (−1 − 1.73i)9-s + (−0.150 + 0.261i)11-s − 0.554·13-s + 0.774·15-s + (0.363 − 0.630i)17-s + (−0.573 − 0.993i)19-s + (−0.312 − 0.541i)23-s + (0.400 − 0.692i)25-s + 1.73·27-s − 1.11·29-s + (0.0898 − 0.155i)31-s + (−0.261 − 0.452i)33-s + (0.410 + 0.711i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.497596 - 0.246672i\)
\(L(\frac12)\) \(\approx\) \(0.497596 - 0.246672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.5 + 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.19654751851798094008329726799, −9.468793693356120918791439686905, −8.805699277936019474710013521056, −7.60070243381425170115784259814, −6.44678880370783215481637996288, −5.46109827093559425374759838335, −4.68275428627776046058713422853, −4.11074067107696475497037067902, −2.72116242473673604830463391839, −0.32878365276115475990596103057, 1.33238404356840450097045312711, 2.51554974559482468729068402250, 3.99753842342516431449554238152, 5.57789199133190892364132919730, 5.95065094172571818152470857173, 7.09248415526312232094920061073, 7.56516224235763748853997438415, 8.375059857497353581552264913768, 9.657421074763476241599338425945, 10.81287541000616260925266606203

Graph of the $Z$-function along the critical line